I'm having a hard time showing that $\displaystyle \lim_{c \to \0} \int_{c}^{1}|\frac{sin(\frac{1}{x})}{x}|dx $ fails to converge absolutely, any insight would be greatly appreciated.
Thanks to both of you for your help. I did try one way before I read this, maybe you can tell me this works...
$\displaystyle \lim_{c\to\0}\int_{c}^{1}|\frac{sin(\frac{1}{x})}{ x}|dx\geqslant \sum_{k = 1}^\infty\int_{\frac{2}{(4k+1)\pi} }^{\frac{2}{(4k-1)\pi} }|\frac{sin(\frac{1}{x})}{x}|dx \geqslant \sum_{k = 1}^\infty\int_{\frac{2}{(4k+1)\pi} }^{\frac{2}{(4k-1)\pi} }\frac{dx}{x}$
...and then just evaluate, I could be way off.